THE  LIBRARY 

OF 

THE  UNIVERSITY 

OF  CALIFORNIA 

RIVERSIDE 

GIFT  OF 

U.S. Naval  Ordnance  Laboratory 

through 
Dr. E.L.Harrington  Estate 


of  Cbicago 


A  REDETERMINATION  OF 
THE  COEFFICIENT  OF  VISCOSITY  OF  AIR 


A   DISSERTATION 

SUBMITTED  TO  THE  FACULTY  OF  THE  OGDEN  SCHOOL  OF  GRADUATE 

SCIENCE  IN  CANDIDACY  FOR  THE  DEGREE  OF 

DOCTOR  OF  PHILOSOPHY 

(DEPARTMENT  OF  PHYSICS) 


BY 

ERTLE   LESLIE   HARRINGTON 


A  Private  Edition 

Distributed  by 
The  University  of  Chicago  Libraries 


Reprinted  from  THE  PHYSICAL  REVIEW,  N.  S.,  Vol.  VIII,  No.  6, 
December,  1916 


A  REDETERMINATION  OF  THE  ABSOLUTE  VALUE  OF  THE 
COEFFICIENT  OF  VISCOSITY  OF  AIR. 


[Reprinted  from  the  PHYSICAL  REVIEW,  N.S..  Vol.  VIII,  No.  6,  December,  1916.] 


A  REDETERM I  NATION  OF  THE  ABSOLUTE  VALUE  OF  THE 
COEFFICIENT  OF  VISCOSITY  OF  AIR. 

BY  ERTLE  LESLIE  HARRINGTON. 

IN  view  of  certain  results  on  the  "Coefficients  of  Slip"  obtained  by  him 
from  his  study  of  the  laws  of  fall  of  small  spheres  in  gases,  Prof. 
Millikan  suggested  to  me  that  I  attempt  to  check  these  results  by  making 
some  measurements  on  the  coefficients  of  viscosity  at  very  low  pressures 
with  the  constant  deflection  apparatus  designed  by  himself  and  Dr. 
Gilchrist.  In  the  course  of  this  work  the  method  was  so  perfected  as  to 
make  it  capable  of  a  precision  apparently  unapproached  heretofore  in 
measurements  on  the  viscosity  of  gases. 

Since  the  knowledge  of  the  exact  value  of  this  coefficient  for  air  is  of 
such  fundamental  importance  in  many  lines  of  physical  research  it 
seemed  worth  while  to  turn  aside  for  the  sake  of  attempting  again  to 
fix  its  value  with  all  the  precision  possible.  This  was  especially  needed 
since  some  question1  has  been  raised  recently  relative  to  the  reliability 
of  the  estimate  made  by  Prof.  Millikan  as  to  the  most  probable  value  of 
this  constant.  From  determinations  made  under  his  direction  by  Gil- 
christ2 by  the  constant  deflection  method,  and  by  Rapp3  by  an  improved 
capillary  tube  method,  taken  in  connection  with  reliable  determinations 
made  elsewhere  and  by  other  methods,  Prof.  Millikan  published4  as  the 
most  probable  value  of  this  constant,  ?;,  at  23°  C.,  .0001824,  and  estimated 
its  uncertainty  at  not  more  than  one  tenth  per  cent. 

In  connection  with  another  problem,  Vogel,5  in  1914,  was  led  to  make 
and  publish  a  summary  of  the  results  of  nearly  all  the  determinations  of 
this  constant  that  had  ever  been  made,  but  he  arrived  at  a  value  about 
.8  per  cent,  higher  than  the  above.  In  this,  however,  he  includes  deter- 
minations which  certainly  involve  gross  error  and  which  are  at  least  very 
far  from  what  is  now  known  to  be  the  approximate  value  of  rj.  In  fact, 
he  lists  values  with  a  total  range  of  nearly  n  per  cent.,  even  in  the  same 
method,  while  observers  with  present  laboratory  methods  and  facilities, 

1  A.  Gille,  Ann.  der  Phys.,  48,  p.  799,  1915. 

2  PHYS.  REV.,  I.,  p.  124,  1913. 

3  PHYS.  REV.,  2,  p.  363,  1913. 

4  Ann.  der  Phys.,  41,  p.  759,  1913. 

6  Ann.  der  Phys.,  43,  p.  1235,  1914. 


VOL.  VIII.l 
No.  6. 


COEFFICIENT   OF   VISCOSITY   OF  AIR. 


739 


using  any  method  whatsoever,  would  admit  no  greater  error  than  a 
fraction  of  a  per  cent.,  hence  the  inclusion  of  such  widely  variant  values 
of  77  renders  more  doubtful  the  validity  of  any  mean  so  obtained. 

The  determination  by  Gilchrist  appears  also  in  this  summary  by 
Vogel,  and  although  he  gives  it  relatively  more  weight  than  any  other 
single  determination  by  any  method,  he  nevertheless  expresses  the  feeling 
that  the  method  and  theory  had  not  as  yet  received  full  development. 
Since  the  publication  of  the  Gilchrist  article  the  method  has  been  used 
by  Timiriazeff,1  but  his  experimental  arrangements  were  such  as  to  make 
it  better  suited  to  relative  than  to  absolute  determinations.  It  therefore 
seemed  important  to  subject  the  method  itself  to  as  critical  a  study  as 
possible  while  making  the  new  determination  of  77. 


APPARATUS. 

The  apparatus  is  the  same  as  that  used  by  Gilchrist  save  for 
pension  and  for  certain  other  features  which 
it  was  necessary  to  modify  in  order  to  adapt 
it  to  running  in  vacuum.  The  diagram  shows 
the  general  arrangement  of  the  apparatus. 
The  outer,  O,  of  two  concentric  brass  cylin- 
ders is  made  to  rotate  with  constant  velocity 
about  the  inner,  7,  which  is  suitably  hung  by 
an  elastic  suspension.  The  viscosity  of  the 
air  produces  a  drag  upon  the  inner  cylinder 
causing  it  to  be  deflected  from  its  position  of 
rest  to  such  an  angle  that  the  restoring  couple 
of  the  suspension  brought  into  play  exactly 
counteracts  the  drag  of  the  air,  the  angle  of 
deflection  being  measured  by  the  usual  mirror 
telescope  and  scale  method. 

The  cylinder  frame,  F,  is  mounted  upon  a 
heavy  steel  plate,  P,  accurately  machined  and 
having  a  raised  rim  in  order  to  provide  a 
mercury  seal  for  the  large  glass  jar,  /,  which 
covers  the  whole.  To  one  side  of  the  plate  is 
drilled  a  hole  to  provide  pump  connections, 
and  in  the  bottom  is  screwed  a  steel  pipe,  Q, 
enclosing  the  driving  shaft,  and  of  such  length 
as  to  serve  as  a  barometer  column,  and  the 
lower  ends  of  pipe,  rod,  and  gearing  were  suit- 
ably constructed  to  dip  into  a  mercury  cup, 

1  Ann.  der  Phys.,  40,  p.  971,  1913. 


the  sus- 


Fig.  1. 


74°  ERTLE  LESLIE   HARRINGTON.  FSECOND 


[SERIES. 


C,  through  which  the  motion  must  be  transmitted.  The  cylinders  are 
carried  by  a  heavy  brass  frame,  F,  provided  with  leveling  screws  and  levels. 
The  outer  cylinder  is  supported  on  a  slightly  conical  bearing,  B,  at  the  bot- 
tom, and  at  the  top  rotates  on  the  tube,  T,  as  an  axle.  The  tube,  T,  at 
the  same  time  supports  the  suspension  head,  A,  and  is  rigidly  held  by 
the  main  frame.  To  eliminate  end  effects  the  inner  cylinder  is  provided 
at  either  end  with  a  guard  cylinder,  G,  of  the  same  radius,  and  rigidly 
held  at  a  small  distance  from  it  and  in  perfect  alignment  with  it  by  three 
brass  posts  which  connect  the  two  guard  cylinders  and  suitable  end  disks. 
The  upper  part  of  this  guard  system  and  the  suspension  tube  above 
mentioned  are  constructed  as  one  piece  so  that  the  upper  guard  cylinder 
is  held  accurately  centered  with  respect  to  the  upper  part  of  the  rotating 
•cylinder.  The  lower  part  rests  on  a  conical  bearing  which  is  a  part  of, 
and  accurately  centered  with  respect  to  the  bottom  of  the  outer  cylinder. 
The  suspension  head,  A,  is  so  constructed  as  to  permit  considerable 
translatory  motion  in  any  direction,  as  well  as  rotatory,  thus  making 
possible  accurate  adjustment.  The  above  apparatus  was  constructed 
with  great  care  and  precision  by  Wm.  Gaertner  &  Co.,  and  to  them 
much  credit  is  therefore  due  for  the  success  of  the  experiment.  Such 
accuracy  in  construction  made  it  possible  to  have  the  guard  cylinders 
very  close  (.025  cm.)  to  the  suspended  cylinder,  and  thereby  completely 
eliminate  the  need  of  end  effect  corrections. 

A  chronograph,  K,  provided  with  extra  weights  serves  the  double 
purpose  of  driving  the  apparatus,  and  also  of  leaving  a  permanent  record 
of  the  speed  of  rotation.  It  was  so  arranged  that  the  cylinder  could  be 
instantly  thrown  out  of  gear  and  the  chronograph  used  independently 
of  the  other  apparatus,  this  being  done  each  time  the  period  of  vibration 
of  the  inner  cylinder  was  determined. 

A  Beckmann  thermometer,  calibrated  with  a  standard  Baudin  ther- 
mometer, and  reading  directly  to  .01  degree  was  hung  beside  the  outer 
cylinder.  A  high-grade  Centigrade  thermometer  graduated  to  .1  degree 
was  hung  beside  it  to  serve  as  a  check.  The  room  itself  was  one  of  the 
constant  temperature  rooms  of  the  Ryerson  laboratory;  a  basement 
room,  with  no  windows,  lined  on.  all  sides,  top  and  bottom,  with  a  5-in. 
layer  of  cork,  and  provided  with  heavy  inner  and  outer  doors.  The 
room  was  efficiently  heated  by  a  new  style  electric  heater  (furnished  by 
the  Lee  Electric  Radiator  Co.)  used  in  connection  with  a  sensitive  thermo- 
static  control,  and  the  air  was  constantly  stirred  by  a  large  fan.  The 
temperature  control  was  such  that  during  any  run  the  temperature 
variation  recorded  by  the  Beckmann  thermometer  was  never  more  than 
a  few  hundred ths  of  a  degree,  and  often  no  change  at  all  was  detected. 


VOL.  VIII.1 
No.  6.        J 


COEFFICIENT   OF    VISCOSITY   OF   AIR. 


741 


The  air  in  the  apparatus  was  kept  dry  by  an  enclosed  dish  of  phos- 
phorus pentoxide. 

CRITICAL  STUDY  OF  APPARATUS  AND  DETERMINATION  OF  DIMENSIONS. 

Inasmuch  as  one  of  the  objects  of  the  experiment  was  to  study  the 

possibilities  of  the  method  as  well  as  to  find  the  absolute  value  of  17, 

considerable  time  was  spent  on  this  phase  of  the  work,  and  w  enever 


Fig.  2. 

possible  various  methods  of  measurement  were  used  for  the  purpose  of 
cross  checking. 

i.  The  Inner  Cylinder. — Three  methods  were  used  to  measure  the 
diameter  of  the  inner  cylinder.  In  order  to  insure  getting  different 
diameters  and  to  entitle  each  different  one  to  equal  weight  in  the  computa- 
tion of  the  mean  a  large  number  of  points  evenly  distributed  over  the 
surface  of  the  cylinder  were  systematically  numbered  and  the  measure- 
ments were  taken  at  these  points.  The  first  method  was  to  place  the 
cylinder  vertically  on  the  bed  of  a  dividing  engine  having  mounted  at  one 


742  ERTLE   LESLIE   HARRINGTON. 

side  with  its  axes  perpendicular  to  the  direction  of  motion  a  short  focus 
telescope  provided  with  cross  hairs.  The  distance  between  tangent 
lines  was  then  determined  from  the  screw  readings.  The  second  method 
was  to  adjust  the  jaws  of  a  large-size  vernier  caliper  to  the  diameter  of 
the  cylinder  and  then  determine  the  perpendicular  distance  between  the 
locked  jaws  by  placing  the  same  on  the  bed  of  the  dividing  engine,  and 
measuring  by  the  usual  method.  The  third  method  was  to  use  a 
micrometer  screw  caliper  of  sufficiently  large  size  to  measure  directly 
the  diameter.  The  last  two  methods  were  by  far  the  more  convenient 
and  yielded  as  accurate  results.  These  two  results  differed  by  only  I 
part  in  13,000,  and  their  mean  agreed  to  Gilchrist's  value  to  within  I 
part  in  6,000,  which  is  a  liberal  estimate  of  the  possible  error  in  this 
dimension.  The  length  was  determined  by  cathetometer  methods,  and 
the  value  found  to  agree  exactly  with  that  given  by  Gilchrist,  and  is 
probably  correct  to  within  I  part  in  6,000. 

2.  The  Outer  Cylinder. — The  most  satisfactory  method  of  measuring 
the  diameter  of  the  outer  cylinder  was  found  to  be  by  filling  it  with  dis- 
tilled water  observing  the  temperature  and  depth,  and  weighing  the  water 
used.  Results  thus  obtained  differed  by  only  I  part  in  15,000,  and  the 
mean  differed  from  that  by  Gilchrist  by  exactly  the  same  amount, 
so  the  error  is  perhaps  no  more  than  I  part  in  8,000. 

The  accuracy  in  the  inner  surface  was  studied  in  this  way :  The  inner 
cylinder,  the  lower  guard  ring,  and  the  posts  were  removed,  leaving  the 
upper  part  of  the  guard  cylinder  system,  which,  as  above  described,  form 
the  axle  for  the  outer  cylinder.  A  heavy  rod  with  one  end  hollowed 
out  to  fit  the  conical  bearing  in  the  bottom  of  the  outer  cylinder  was 
inserted  in  the  opening  and  brought  to  rest  on  the  bearing.  This  rod 
carried  a  lever  arm  suitably  curved  to  rest  against  the  wall  of  the  cylinder, 
capable  of  being  adjusted  to  different  heights,  and  bearing  at  the  fulcrum 
a  mirror.  A  suitable  telescope  with  vertical  scale  placed  at  a  distance 
of  nearly  three  meters  made  it  possible  to  detect  the  slightest  deviation 
from  constancy  in  the  radius  or  perfection  in  symmetry.  In  doing  this 
the  lever  was  held  lightly  against  the  wall  by  a  weight,  and  the  cylinder 
slowly  rotated  at  a  constant  speed.  From  the  readings  of  the  observer 
at  the  telescope  and  a  measurement  of  all  distances  concerned,  any 
variations  could  be  quantitatively  determined.  It  is  important  to  note 
that  this  method  not  only  tests  the  accuracy  of  the  inner  surface  but 
tests  also  the  symmetry  of  this  surface  about  the  same  bearing  that  is  to 
carry  and  hold  in  position  the  inner  cylinder  system.  The  results  of  this 
test  were  very  satisfactory,  inasmuch  as  the  variations  were  of  the  order 
of  i  part  in  2,500  and  of  such  nature  as  to  be  self-counteracting  in  their 


NoL6VIIL]  COEFFICIENT  OF    VISCOSITY  OF  AIR.  743 

effect  on  the  final  results,  which  were  therefore  affected  probably  less 
than  i  part  in  5,000. 

3.  Moment  of  Inertia  of  the  Inner  Cylinder.  —  This  was  found  as  usual 
by  determining  the  period  of  vibration  of  the  cylinder  alone,  and  then 
when  a  known  moment  of  inertia  was  added.  The  cylinder  was  sus- 
pended by  piano  wire  and  the  cylindrical  surface  made  plumb  and 
symmetrical  about  the  axis  by  suitable  adjustment  of  the  three  support 
bars  which  connect  the  cylinder  with  the  suspension  clamp,  and  by  use 
of  small  weights  fastened  to  the  upper  supporting  vane  of  the  cylinder. 
The  heavy,  double  support  of  the  clamp  from  which  the  cylinder  was 
suspended  rested  on  a  stone  bench,  and  although  the  room  was  apparently 
free  from  air  currents  the  cylinder  was  surrounded  by  a  much  larger  one 
in  order  to  insure  entire  absence  of  them.  The  passage  through  the  zero 
position  of  the  cylinder  was  indicated  by  the  flash  of  a  light  into  a  tele- 
scope at  the  opposite  side  of  the  room,  and  an  electric  key  enabled  the 
operator  to  accurately  register  this  time  on  the  chronograph.  The  time 
divisions  were  marked  by  impulses  from  the  standard  laboratory  clock 
and  involved  no  appreciable  error  whatever,  and  the  divisions  could  be 
read  to  one  one  hundredth  of  a  second.  By  taking  the  period  from  a 
rather  long  run  thus  measured,  the  individual  runs  varied  from  the  mean 
for  any  suspension  by  an  amount  of  the  order  of  I  part  in  10,000,  which 
may  be  considered  the  probable  error  from  this  source.  The  added 
known  inertia  consisted  of  a  bar  and  ring,  accurately  machined,  and 
placed  on  the  cylinder  symmetrically.  The  dimensions  of  the  ring  and 
bar  were  determined  on  the  dividing  engine,  and  their  weights  by  the 
analytical  balance.  The  inertia  of  the  combined  ring  and  bar  was 
computed  by  the  usual  formula  derived  for  such  bodies,  and  the  result 
agreed  to  I  part  in  8,000  with  that  obtained  by  Gilchrist,  this  ratio 
probably  representing  the  accuracy  of  this  determination.  No  faulty 
adjustment  to  symmetry  about  the  axis  of  suspension  could  have  been 
constant  to  the  different  series  of  runs,  since  the  ring  and  bar  were 
frequently  removed  and  replaced. 

Having  the  two  periods  and  the  one  known  inertia,  the  moment  of 
inertia  of  the  inner  cylinder  was  computed  from  the  usual  formula, 


_ 

r2        / 

The  average  departure  of  the  various  results  from  the  mean  for  the 
moment  of  inertia  of  the  inner  cylinder  obtained  from  the  different 
suspensions  used  was  I  part  in  5,000  and  the  mean  differed  from  that 
obtained  by  Gilchrist  by  exactly  the  same  amount. 


744  ERTLE  LESLIE  HARRINGTON. 

DIMENSIONS. 

Dimensions  that  are  not  involved  in  the  computation  of  the  results  are 
here  included  for  descriptive  purposes,  but  only  approximate  values  for 
such  are  given. 

Vertical  distance  from  base  of  chronograph  to  extreme  top  of  cover      185  cm. 

Diameter  and  height  of  glass  jar,  respectively 28  and  62   cm. 

Length  of  outer  cylinder 46 

Length  of  suspension 23.5 

Length  of  each  guard  cylinder 10 

Length  of  inner  cylinder 24.88 

Distance  between  guard  cylinders 24.93 

Weight  of  inner  cylinder 321    g. 

Radius  of  outer  cylinder 6.06317  cm. 

Radius  of  inner  cylinder 5.341 16 

Moment  of  inertia  of  inner  cylinder  (experimentally  determined) .  .    7,617.3 

ADJUSTMENT  OF  INSTRUMENT. 

Before  being  placed  in  the  instrument  the  inner  cylinder  was  again 
tested  for  symmetry  and  perpendicularity  with  the  same  methods  and 
precautions  used  preliminary  to  the  determination  of  its  moment  of 
inertia.  In  this  three  plumb  bobs  instead  of  one  were  used  for  the 
purpose  of  expediting  matters  and  permitting  simultaneous  observations 
in  three  directions  without  in  any  way  disturbing  the  cylinder  or  plumb 
bobs. 

The  lower  end  of  the  outer  cylinder  being  in  place,  the  inner  cylinder 
and  the  guard  cylinder  system  were  put  into  position,  fastened  rigidly 
by  the  top  brace,  and  after  suspending  the  inner  cylinder  the  suspension 
head  screws  and  the  leveling  screws  of  the  base  of  the  instrument  were  so 
manipulated  as  to  bring  the  surface  of  the  guard  cylinders  into  alignment 
with  the  inner  cylinder  as  perfectly  as  possible  without  the  use  of  a 
telescope.  The  outer  cylinder  was  then  put  into  position,  the  suspension 
tube  again  clamped,  and  the  deflection  of  the  inner  cylinder  noted  as  the 
outer  cylinder  was  rotated  at  the  speed  to  be  used  later  in  determinations. 
The  suspension  head  was  then  rotated  until  the  zero  position  was  brought 
as  much  to  one  side  as  the  deflection  position  was  to  the  other  side  of 
the  telescope  which  was  in  the  middle  of,  and  perpendicular  to  the 
scale.  Three  small  support  screws  in  the  base  of  the  instrument  were 
then  brought  just  to  touch  the  bottom  of  the  outer  cylinder  in  order  to 
hold  it  in  its  exact  position  during  the  final  adjustment.  The  outer 
cylinder  was  loosened  from  its  bottom  and  removed  and  the  top  brace 
replaced  and  fastened  firmly  in  position.  The  final,  and  more  careful 
adjustment  of  the  guard  cylinder  system  to  alignment  with  the  inner 
cylinder  was  made  by  means  of  a  short  focus  telescope.  With  a  suitable 


JJ°L-6yiIL]  COEFFICIENT   OF   VISCOSITY   OF  AIR.  745 

clamp  system  attached  to  the  frame  the  outer  cylinder  was  replaced 
without  at  any  stage  of  the  process  inclining  or  otherwise  disturbing  the 
inner  cylinder.  The  preliminary  adjustment  to  zero  position  made  un- 
necessary the  turning  of  either  the  guard  cylinder  system  or  the  suspen- 
sion head  after  this  final  adjustment,  and  thereby  eliminated  any  error 
that  might  have  come  from  a  lack  of  perfect  coincidence  of  the  suspension 
with  the  axis  of  the  suspension  head  collar.  The  use  of  the  support  screws 
for  the  base  of  the  outer  cylinder  holds  it,  and  therefore  the  lower  end  of 
the  inner  cylinder  system,  in  precisely  the  position  they  occupy  during 
a  run.  Keeping  the  guard  cylinder  system  in  place  while  replacing  the 
outer  cylinder  eliminates  the  question  as  to  whether  it  returns  to  the 
exact  position  it  had  during  adjustment.  With  these  precautions  it 
would  seem  unlikely  that  any  appreciable  error  could  arise  from  faulty 
adjustment,  and  the  results  later  given  involving  results  obtained  before 
and  after  dissembling  and  readjusting  furnish  convincing  evidence  of 
the  absence  of  such  error. 

TRIAL  RUNS  AND  A  STUDY  OF  FACTORS  AFFECTING  RESULTS. 
As  will  appear  later,  the  value  of  rj  is  computed  from  the  time  of  rota- 
tion, the  period  of  vibration  of  the  inner  cylinder,  the  deflection,  the 
distance  to  the  scale,  and  the  temperature  observations.  The  method  of 
taking  these  observations  was  this:  The  zero  position  was  read  when  the 
cylinder  was  at  rest  and  checked  by  causing  the  cylinder  to  make  small 
vibrations  (about  I  cm.  as  seen  on  the  scale)  and  determining  the  zero 
position  as  is  done  in  using  balances.  This  was  done  as  a  precaution 
against  any  error  due  to  sticking,  although  experience  showed  this  step 
really  unnecessary.  The  outer  cylinder  was  then  put  into  rotation,  the 
speed  at  first  being  modified  by  a  brake  attachment  to  the  chronograph 
which  could  be  operated  from  the  position  of  the  telescope  in  order  to 
bring  the  inner  cylinder  quickly  to  near  rest  in  its  deflected  position. 
As  soon  as  a  steady  state  was  attained  the  stylus  was  lowered  on  to  the 
waxed  paper  of  the  chronograph  drum  and  the  rotation  continued  until 
the  stylus  had  traveled  the  length  of  the  drum,  which  meant,  with  the 
speed  used,  an  interval  of  about  13  min.  In  practice  the  inner  cylinder 
was  allowed  to  vibrate  through  two  or  three  millimeters  as  seen  on  the 
scale,  readings  being  taken  at  short  intervals;  this  plan  giving  a  large 
number  of  independent  readings,  eliminating  the  barely  possible  source 
of  error  due  to  sticking,  and  affecting  a  great  saving  of  time  that  would 
otherwise  be  required  to  bring  the  cylinder  to  absolute  rest,  inasmuch  as 
the  period  of  vibration  was  very  long,  and  the  damping  very  small. 
At  frequent  intervals  during  this  period  the  temperature  was  read  on 


746  ERTLE   LESLIE   HARRINGTON.  [SEMES! 

the  Beckmann  thermometer.  At  the  close  of  the  run  the  cylinder  was 
slowly  let  back  and  its  zero  position  again  checked.  Now  the  outer 
cylinder  was  given  a  rotation  sufficient  to  set  the  inner  cylinder  in  vibra- 
tion, and  then  thrown  out  of  gear,  thus  allowing  the  chronograph  to  be 
used  merely  as  such,  whereupon  the  period  of  vibration  was  taken  over  a 
period  of  about  45  min.  To  do  this  an  electric  key  at  the  position  of 
the  telescope  enabled  the  operator  to  make  accurate  chronograph  record 
of  the  passage  through  the  zero  position,  the  first  five  and  the  last  five 
being  recorded  in  order  to  provide  means  of  cross  checking  and  thus 
insure  absence  of  appreciable  error  from  this  source.  The  periods  thus 
obtained  were  probably  accurate  to  within  I  part  in  8,000. 

The  scale  on  which  the  deflections  were  read  was  a  carefully  selected 
straight  meter  stick  tested  with  a  standard  metric  steel  scale.  The 
magnification  of  the  telescope  was  such  that  .1  mm.  could  be  read  easily 
and  since  the  deflections  were  of  the  order  of  600  mm.  the  readings  were 
probably  correct  to  within  I  part  in  6,000.  A  steel  tape  was  used  to  set 
the  ends  of  the  scale  equidistant  fiom  the  mirror  and  to  determine 
its  perpendicular  distance  from  the  mirror.  The  error  in  this  was  prob- 
ably not  greater  than  I  part  in  6,oob. 

Irregularities  in  the  speed  of  the  chronograph  might  offer  a  source  of 
error,  not  on  account  of  uncertainty  as  to  what  the  speed  is,  for  that  is 
obtainable  directly  from  the  record,  but  on  account  of  the  resulting 
unsteadiness  of  the  deflection.  Fortunately  it  was  found  that  the  chrono- 
graph drove  the  apparatus  at  a  very  constant  speed,  though  of  course 
not  perfectly  so,  owing  perhaps  to  the  irregularities  in  the  friction,  or  a 
lack  of  perfection  in  the  gearing.  However,  a  small  auxiliary  weight, 
at  the  side  of  the  operator,  brought  into  series  with  the  main  weight  by 
two  pulleys,  enabled  him  after  some  experience  to  almost  completely 
neutralize  any  such  irregularities,  thereby  reducing  the  error  from  this 
source  to  probably  I  part  in  5,000. 

The  high  consistency  in  the  early  trials  in  the  results  for  any  suspension 
indicated  a  satisfactory  control  in  all  the  above  sources  of  error,  but  it 
was  found  that  variations  in  the  suspension  produced  rather  great 
variations  in  the  results.  In  view  of  the  experience  of  Gilchrist  the 
bifilar  form  of  suspension  was  tried  at  first,  but  on  account  of  the  rather 
great  weight  (321  g.)  of  the  inner  cylinder,  and  its  small  distance  (.25 
mm.)  from  the  guard  cylinders  which  permitted  no  sag,  it  was  not  possible 
to  use  silk  fibers  which  permit,  perhaps,  the  closest  approach  to  the  true 
bifilar  type.  Metallic  ribbons  were  therefore  used,  but  after  two  months 
of  experience  with  them  they  were  wholly  discarded,  since  it  was  not 
found  possible  to  entirely  eliminate  the  effect  of  a  change  in  the  ribbon 


No"6VIIL]  COEFFICIENT  OF    VISCOSITY   OF  AIR.  747 

or  even  of  a  mere  change  in  the  separation  of  the  strands  or  in  the  manner 
of  clamping,  upon  the  results  obtained.  The  trouble  no  doubt  lies  par- 
tially in  the  rather  great  discrepancy  between  such  a  bifilar  and  those 
ordinarily  treated  theoretically  since  the  deflection  brings  into  play  not 
only  a  restoring  couple  due  to  the  slight  raise  in  the  weight,  but  also  a 
restoring  couple  due  to  the  twist  in  the  strands  themselves.  Such  a 
suspension  is  therefore  a  sort  of  hybrid  between  the  bifilar  and  the 
ordinary  elastic  unifilar  suspension.  The  greatest  error,  however,  prob- 
ably comes  from  the  fact  that  the  ribbons  have  widths  of  the  same  order 
of  magnitude  as  the  separation  of  the  strands  which  makes  it  quite  likely 
that  as  deflection  occurs  the  two  edges  of  either  strand  may  assume 
varying  portions  of  the  load,  thereby  causing  a  virtual  change  in  the 
distance  between  the  strands.  If  the  strands  be  clamped  at  both  ends 
it  is  unlikely  that  the  load  is  equalized  between  the  two  strands,  and  if 
instead  the  strand  be  simply  looped  about  a  pin  at  one  end  in  order  to 
permit  constant  equalization  of  the  load  there  is  the  possibility  of  error 
due  to  a  rolling  of  the  strands  about  the  pin  as  the  deflection  occurs. 
Unrolled  phosphor  brojize  wire  in  place  of  the  ribbon  was  even  less 
satisfactory  owing  to  the  residual  coil  and  the  consequent  drift,  nor  was 
either  found  satisfactory  later  as  a  unifilar  suspension.  In  fact  the 
writer  was  led  to  conclude  that  little  dependence  could  be  placed  on  phos- 
phor bronze  where  precise  results  are  expected.  Quartz  fibers  were  tried, 
but  it  was  not  found  possible  to  obtain  fibers  coarse  enough  to  support 
the  rather  great  weight  and  at  the  same  time  fine  enough  to  give  sufficient 
deflections.  The  smallest  piano  steel  wire  obtainable  was  much  too  stiff, 
but  it  was  found  possible  by  reducing  the  size  of  the  smallest  obtained, 
by  very  carefully  rubbing  with  fine  emery  paper,  to  secure  a  sufficiently 
large  deflection  and  yet  retain  adequate  tensile  strength.  After  the 
adoption  of  this  plan  no  further  suspension  troubles  were  experienced. 
A  number  of  such  were  made,  and  in  all  cases  there  was  a  good  zero 
return  and  a  satisfactory  absence  of  drift.  Moreover,  different  suspen- 
sions, though  differing  greatly  in  stiffness,  yielded  entirely  concordant 
results.  Later  some  samples  of  the  uncoiled  stock  from  which  the  hair 
springs  of  watches  are  made  were  furnished  by  the  Elgin  Watch  Company 
and  found  to  have  the  proper  range  of  stiffness. 

As  will  be  seen  later  the  torsion  constant  of  the  suspension  is  expressed 
in  terms  of  the  inertia  of  the  inner  cylinder  and  the  period  of  vibration. 
From  the  observed  damping,  and  the  theoretical  relation  between  the 
magnitude  of  damping  and  the  effect  on  the  period  as  given,  for  example, 
by  Helmholtz,  it  was  calculated  that  the  total  effect  due  to  the  damping 
factors  would  be  of  the  order  of  I  part  in  10,000,  or  quite  inappreciable. 


748  ERTLE  LESLIE  HARRINGTON. 

Moreover,  by  experiment  no  effect  on  the  period  could  be  observed  when 
the  outer  cylinder  was  removed  and  the  guard  cylinders  were  separated 
many  times  as  far  from  the  vibrating  cylinder.  But,  although  such 
factors  as  viscosity  involved  above  do  not  appreciably  affect  the  period, 
the  fact  must  not  be  overlooked  that  the  vibrating  cylinder  does  carry 
the  air  with  it,  and  the  moment  of  inertia  of  this  air  must  be  taken  into 
account.  This  point  was  first  called  to  my  attention  by  Dr.  Lunn. 
The  periods  taken  in  vacua  were  actually  found  to  be  i  part  in  750  less 
than  the  periods  taken  at  ordinary  pressures,  so  the  data  given  make  due 
allowance  for  this  effect. 

COMPUTATION  OF  RESULTS. 

For  the  calculation  of  the  results  the  well-known  and  very  simple 
formula1  was  used  in  this  form  : 

-  a2) 


where  77  is  the  coefficient  of  viscosity,  I  the  moment  of  inertia  of  the  inner 
cylinder,  a  and  b  the  radii  of  the  inner  and  outer  cylinders  respectively, 
I  the  length  of  the  inner  cylinder,  <f>  the  angular  deflection  of  the  inner 
cylinder,  T  the  period  of  vibration  of  the  inner  cylinder,  and  o>  the 
constant  angular  velocity  of  the  outer  cylinder.  If  the  period  of  rotation 
of  the  outer  cylinder,  /,  be  substituted  for  27r/w,  and  a  constant,  K,  for 
the  product  [I(b2  —  a2)]/(2a2&2/)  (having  here  a  numerical  value  of 
1.20188)  the  formula  becomes: 

tK<f>  tK  s 

77  =  —     or     i,  =  —tan  l—  , 

where  5  is  the  deflection  as  read  on  a  straight  scale,  and  d  the  distance 
of  the  scale  from  the  mirror.  The  latter  form  was  used  in  all  computa- 
tion. A  development  of  the  above  formula  involving  a  more  general 
treatment  which  considers  the  coefficient  of  slip  will  be  given  in  a  paper, 
following  this,  which  will  consider  the  problem  of  viscosity  at  the  low 
pressures  where  the  effect  of  slip  becomes  appreciable.  In  apparatus 
of  the  dimensions  here  used  the  correction  for  slip  at  ordinary  pressures 
amounts  to  about  2  parts  in  100,000. 

All  determinations  were  made  at  temperatures  in  the  neighborhood  of 
23°  C.,  and  the  values  for  77  reduced  to  that  temperature  by  the  use  of 
the  formula  suggested  by  Prof.  Millikan  (1.  c.), 

1723  =  779  +  .000000493(23  -  0), 
where  779  is  the  value  of  the  viscosity  coefficient  obtained  at  0°  C.     This 

1  See  Poynting  and  Thompson,  Properties  of  Matter,  p.  213. 


VOL.  VIII.l 
No.  6. 


COEFFICIENT   OF    VISCOSITY  OF  AIR. 


749 


simple  formula,  found  satisfactory  through  the  range  mentioned  by  him 
must  certainly  hold  in  the  small  ranges  here  involved,  which  are,  with 
but  two  exceptions,  less  than  one  degree. 

RESULTS. 

The  following  data  show  the  results  of  thirty-one  determinations  of 
77  made  at  various  times  during  about  three  months,  and  involve  the  use 
of  six  different  suspensions.  Suspensions  C  and  D  were  really  the  same 
suspension  under  different  physical  conditions.  In  most  cases  during 
the  series  of  runs  for  a  given  suspension  the  apparatus  was  taken  apart 
and  readjusted  in  order  to  make  sure  there  was  no  error  of  adjustment. 


Sus. 

«/(Cm.). 

j. 

e. 

t. 

T. 

1>H  X  1C*. 

A 

200.0 

40.21 

23.97 

30.059 

140.70 

1,823.7 

A 

200.0 

40.26 

24.45 

30.020 

140.61 

1,823.5 

A 

200.0 

40.06 

23.06 

30.018 

140.62 

1,820.9 

A 

200.0 

40.05 

23.31 

30.085 

140.62 

1,823.3 

A 

199.7 

40.025 

23.18 

30.000 

140.58 

1,821.4 

B 

200.2 

35.81 

23.61 

30.014 

132.81 

1,821.2 

B 

200.2 

35.665 

22.88 

30.114 

132.79 

1,824.1 

B 

200.2 

35.73 

23.14 

30.031 

132.79 

1,821.0 

B 

200.2 

35.837 

23.43 

29.971 

132.80 

1,821.2 

B 

200.2 

35.80 

23.46 

30.032 

132.80 

1,822.8 

C 

200.8 

60.51 

22.88 

30.090 

172.13 

1,825.9 

C 

200.8 

60.67 

23.055 

29.940 

172.14 

1,820.5 

C 

200.8 

61.064 

22.93 

29.840 

172.21 

1,825.1 

C 

200.8 

60.624 

23.165 

30.014 

172.18 

1,822.2 

C 

200.8 

60.604 

22.84 

29.962 

172.13 

1,821.1 

c 

200.8 

60.68 

23.09 

29.936 

172.13 

1,820.8 

C 

200.8 

60.73 

22.91 

29.940 

172.13 

1,823.1 

C 

200.8 

60.77 

23.19 

29.907 

172.14 

1,821.0 

C 

200.8 

60.75 

22.90 

29.895 

172.14 

1,820.9 

C 

200.6 

60.905 

24.10 

29.912 

172.13 

1,822.7 

C 

200.6 

60.54 

23.00 

29.982 

172.10 

1,822.0 

D 

200.7 

-     61.154 

22.97 

30.014 

172.98 

1,822.9 

D 

200.7 

61.216 

22.99 

29.954 

172.95 

1,821.6 

D 

200.7 

61.07 

23.22 

30.075 

172.96 

1,823.2 

E 

200.85 

51.66 

23.27 

30.178 

159.39 

1,824.8 

E 

200.85 

51.168 

23.28 

29.888 

159.39 

1,824.7 

F 

200.9 

63.34 

23.16 

29.946 

175.62 

1,823.8 

F 

200.9 

62.988 

23.26 

30.087 

175.60 

1,822.3 

F 

200.9 

62.957 

23.11 

30.103 

175.56 

1,824.0 

F 

200.9 

62.974 

23.10 

30.066 

175.58 

1,821.9 

F 

200.9 

63.093 

23.19 

30.008 

175.55 

1,821.9 

Mean  value  of  r,  at  23°  C.  =  1822.6  X  10^. 


750  ERTLE   LESLIE   HARRINGTON. 

Moreover,  the  runs  show  such  variations  in  the  various  factors  involved 
as  to  make  each  determination  an  independent  one,  and  no  determination 
made  with  satisfactory  control  in  the  manipulation  of  all  steps  was 
discarded. 

The  strongest  evidence  of  the  advantages  of  this  method  for  the  deter- 
mination of  77  lies  in  the  remarkable  consistency  of  the  results  here  shown. 
If  from  the  individual  deviations  from  the  above  mean  one  computes 
the  probable  error  by  the  usual  least  square  method  the  result  is  found 
to  be  .19  or  i  part  in  9,600.  Moreover,  it  should  be  emphasized  that  this 
really  includes  every  source  of  error  except  those  involved  in  the  instru- 
ment constant,  K.  The  probable  error  for  each  of  the  various  determina- 
tions involved  in  this  constant  has  been  given  above  in  detail  and  if  the 
probable  error  in  this  constant  be  computed  by  the  same  method  as 
above  it  is  found  to  be  1.9  parts  in  5,000.  The  combined  or  total  error 
would  be  by  this  method  of  calculation  only  I  part  in  2,500  or  .04  per 
cent.  Moreover,  if  the  means  for  the  different  suspensions  be  compared 
it  is  found  that  the  maximum  variation  from  the  above  mean  is  less  than 
.03  per  cent,  with  the  one  exception  of  suspension  £,  a  watch-spring 
suspension  the  use  of  which  was  discontinued  on  account  of  its  tendency 
to  drift.  Considering  these  two  striking  results,  and  making  any  reason- 
able allowance  for  any  unreliability  in  the  least  square  method  of  com- 
puting errors  or  of  the  estimates  made  of  any  individual  probable  error 
it  seems  entirely  justifiable  to  claim  that  the  above  mean  is  correct  to 
within  less  than  .1  per  cent,  of  the  true  value  of  t\  at  the  temperature 
considered. 

A  comparison  of  the  consistency  of  these  results  with  the  lack  of  con- 
sistency of  the  results  obtained  by  any  other  method,  more  especially 
by  the  capillary  tube  method,  shows  the  marked  superiority  of  this 
method.  The  uncertainties  of  the  capillary  tube  method  need  not  be 
considered  here  inasmuch  as  they  are  well  known  and  have  been  dis- 
cussed by  Prof.  Millikan,1  by  Fisher,2  Vogel,3  and  others.  Only  a  few 
points  of  contrast  need  be  mentioned;  the  capillary  tube  method  is 
based  on  incomplete  theory  since  it  does  not  consider  the  effect  of  the 
radial  component  of  the  velocity  and  other  uncertainties  due  to  the 
increase  in  the  volume  of  the  gas  as  it  passes  along  the  tube,  the  general 
end  effects  in  addition  to  the  question  of  the  effect  of  irregularities  in  the 
bore  upon  the  stream  lines,  the  difficulties  in  getting  the  exact  pressures, 
and  above  all,  the  impossibility  of  getting  perfect  tubes  of  the  small  radii 
usually  employed,  and  the  great  difficulty  of  subjecting  any  tube  selected 
to  accurate  examination  as  to  uniformity,  circularity,  and  even  as  to 

1  L.  c.  *  PHYS.  REV.,  29,  p.  147,  1909.  *  L.  c. 


NoL6        ]  COEFFICIENT   OF    VISCOSITY   OF  AIR.  751 

the  value  of  the  radius  itself,  which  enters,  it  should  be  recalled,  in  the 
fourth  power.  The  wide  variation  in  the  results  obtained  by  this  method 
by  different  observers,  and  even  by  the  same  observer  with  different 
tubes,  is  ample  proof  that  these  uncertainties  exist.  On  the  other  hand, 
with  the  method  described  above  the  theory  is  complete,  there  are  no 
end  corrections  involved,  no  expansion  takes  place,  every  step  in  the 
construction  of  the  cylinders  is  subject  to  control,  the  dimensions  are  so 
great  that  they  may  be  determined  with  great  accuracy,  and  every  portion 
of  the  apparatus  is  subject  to  minute  study  for  irregularities,  and  even 
should  such  exist,  their  effect  would  be  far  less  serious  than  in  the  case 
of  the  other  method.  As  to  consistency  with  other  observers  we  are 
essentially  limited  to  the  value  obtained  by  Gilchrist  who  first  used  the 
method  and  made  the  claim  that  his  result  contained  not  more  than  .2 
per  cent,  error.  The  value  here  obtained  differs  from  his  by  less  than 
that  amount.  That  his  values  fluctuate  through  a  greater  range  than 
the  range  here  obtained  is  without  doubt  due  principally  to  the  suspension 
troubles  mentioned  above  which  were  here  so  largely  eliminated,  for 
every  determination  of  his  of  any  instrument  constant  that  could  at  this 
time  be  checked  was  most  critically  examined,  and  no  one  of  them  found 
to  differ  by  more  than  i  part  in  6,000  from  the  value  here  given. 

It  is  interesting  to  note  that  Rapp,1  who  perhaps  came  more  nearly 
completely  eliminating  the  errors  in  the  capillary  tube  method  than 
anyone  else,  and  who  used  a  large  number  of  tubes,  is  practically  identical 
with  the  result  here  obtained,  as  is  also  that  obtained  by  Hogg,2  who  used 
an  oscillation  method.  The  result  obtained  by  Grindley  and  Gibson,3 
using  still  a  different  plan  differs  only  by  about  .03  per  cent.  More 
significant  still  is  the  fact  that  the  value  published  by  Prof.  Millikan  as 
correct  to  within  .1  per  cent,  is  less  than  .08  per  cent,  above  the  value 
here  obtained,  and  as  his  value  was  based  on  perhaps  the  most  accurate 
determinations  by  five  different  methods,  it  would  seem  that  the  above 
claim  that  the  result  here  obtained  is  within  .1  per  cent,  of  the  true  value 
is  well  founded,  since  the  above  mentioned  values  all  lie  well  within 
this  limit. 

In  conclusion  the  writer  wishes  to  acknowledge  his  gratitude  to  Prof. 
Millikan,  who  suggested  the  problem  and  maintained  such  constant  and 
helpful  interest  in  the  research  during  its  progress,  and  to  Prof.  Michelson, 
the  head  of  the  department,  for  various  helpful  suggestions. 
RYERSON  PHYSICAL  LABORATORY, 
UNIVERSITY  OF  CHICAGO, 
June,  1916. 

iL.  c. 

2  Am.  Acad.  Proc.,  40,  18,  p.  611,  1905. 

*  Proc.  Roy.  Soc.,  A,  80,  p.  114,  1908. 


[Reprinted  from  THE  PHYSICAL  REVIEW,  Vol.  21,  No.  3,  March,  1923.] 


A     DETERMINATION     BY    THE    CONSTANT     DEFLECTION 

METHOD  OF  THE  VALUE  OF  THE  COEFFICIENT  OF 

SLIP    FOR    ROUGH    AND    FOR    SMOOTH 

SURFACES   IN  AIR. 

BY  LELAND  JOHNSON  STACY. 

ABSTRACT. 

Coefficient  of  slip  for  rough  and  smooth  surfaces  in  air,  determined  by  the 
constant  deflection  method. — The  apparent  coefficient  of  viscosity  measured 
by  this  exceptionally  precise  method  does  not  come  out  constant  for  the  lower 
pressures  unless  correction  is  made  for  the  slip  at  the  surfaces.  The  relation 
»7p'(i  +  k£p)  =  rj  =  constant,  between  rjp',  the  apparent  viscosity  coefficient, 
and  fp,  the  coefficient  of  slip,  enables  fp  to  be  determined  from  measurements  of 
r)pf  at  pressures  of  0.2  mm  or  lower.  The  apparatus  included  a  vacuum- 
tight  chamber  inside  which  a  cylinder  of  radius  5.341  cm  was  suspended  by  a 
steel  wire  concentric  with  a  cylinder  of  radius  6.063  cm  which  could  be  rotated 
at  a  constant  slow  rate  so  as  to  cause  a  steady  deflection  of  the  inner  cylinder. 
The  accuracy  of  this  method  of  measuring  rjp'  is  so  great  that  the  values  of 
frs  =  fp£/76  all  lie  within  ±  4  per  cent  of  the  mean.  The  chief  difficulty 
was  in  keeping  the  air  pure  because  of  the  gradual  evolution  of  gas,  probably 
hydrogen,  inside  the  apparatus;  but  by  taking  observations  only  shortly 
after  evacuation  this  effect  was  avoided.  For  brass  surfaces,  f,  reduced  to 
23°  and  76  cm,  came  out  66.15  X  io~7  which  is  practically  the  theoretical 
minimum  deduced  by  Millikan  for  a  completely  diffusing  surface,  65.9  X  io~7. 
For  surfaces  coated  with  shellac,  the  coefficient  was  found  to  be  97  X  io~7  for  a 
fresh  surface,  in  agreement  with  the  value  obtained  by  Lee  from  droplet 
measurements,  but  it  decreased  steadily  with  time,  presumably  because  of  a 
roughening  due  to  oxidation,  falling  in  two  months  to  within  3  per  cent  of  the 
theoretical  minimum.  The  early  part  of  this  work  was  done  in  collaboration 
with  E.  L.  Harrington. 

Coefficient  of  viscosity  of  air  at  o.i  mm  is  the  same  as  at  atmospheric  pres- 
sure when  correction  is  made  for  the  slip  effect.  The  constancy  of  the  values 
obtained  for  f  provides  new  evidence  that  the  coefficient  is  independent  of 
the  pressure. 


T 


I.   HISTORICAL  DEVELOPMENT  OF  THE  IDEA  OF  SLIP. 

HE  theory  that  the  coefficient  of  viscosity  of  a  gas  should  be  inde- 
pendent of  the  pressure  was  first  deduced  by  Maxwell  *  from  a 
consideration  of  the  internal  friction  of  molecules  assumed  to  be  rigid 
spheres.  He  first  deduced  the  relation 

(i)  77  =  pel, 

where  17  =  coefficient  of  viscosity;    p  =  density  of  the  gas;    c  =  mean 
1  Phil.  Mag.,  1860,  Vol.  19,  p.  31. 


240  LELAND   JOHNSON  STACY. 

,  molecular  velocity;  /  =  mean  free  path  of  gas  molecule.  Since  the 
density  is  directly  proportional  to  the  pressure,  while  the  mean  free  path 
is  inversely  proportional  to  the  same  quantity,  the  product  pi,  and  there- 
fore 17,  should  be  independent  of  the  pressure.1  In  a  later  paper,2  Maxwell 
reported  some  experimental  tests  of  his  theory  for  pressures  ranging 
from  30  in.  to  0.5  in.  of  mercury.  His  apparatus  consisted  of  a  torsion 
pendulum  of  three  plane-parallel  plates  suspended  by  an  elastic  fiber 
between  four  fixed  plane-parallel  plates.  The  whole  system  was  enclosed 
in  an  airtight  vessel  and  the  logarithmic  decrement  of  the  oscillations 
was  observed  for  different  pressures.  He  found  no  observable  change  in  the 
decrement,  which  is  a  measure  of  the  viscosity,  within  the  pressure  range 
studied. 

The  same  relation  (i)  was  derived  later  by  O.  E.  Meyer.3  By  an 
experimental  arrangement  similar  to  Maxwell's  he  checked  the  theoretical 
deductions  for  the  same  range  of  pressures.  Results  at  pressures  below 
1/60  atmosphere  showed  a  falling  off  of  the  viscosity  coefficient,  which 
he  later  ascribed  to  the  fact  that,  in  the  theory  of  the  experimental 
method,  the  external  friction  (e)  has  been  considered  infinitely  large  in 
comparison  with  the  internal  friction  or  viscosity.  Thus  he  introduced 
into  the  Kinetic  Theory  of  Gases  the  slip  coefficient  f  =  rj/e,  which 
Helmholtz  4  had  previously  defined  for  liquids. 

II.   DETERMINATIONS  OF  THE  COEFFICIENT  OF  SLIP  IN  AIR. 

The  first  experimental  determination  of  the  value  of  the  coefficient  of 
slip  was  made  by  Kundt  and  Warburg  5  who  used  the  capillary  tube 
method.  Their  results  were: 


Tube  No.    j       Pressure.  Temperature. 


I  

33.8  mm 

15°  c 

.00017 

76  X  io~7 

2  

39-0  ' 

.00016 

82  X  io~7 

2  

33-8  " 

.00018 

80  X  io~7 

Mean  

79  X  io~7 

_. 

The  mean  of  their  values  at  15°  C  and  76  cm  is  about  79  X  io~7. 
Millikan  in  a  preceding  article  has  reduced  this  to  23°  C,  getting  f76  at 
23°  C  =  82  X  io-7. 

1  Using  Stokes'  value  Vij/p  =  .116  for  air,  Maxwell  made  the  first  calculation  of  the 
mean  free  path  of  a  gas  molecule. 

2  Phil.  Trans.,  1866,  Vol.  156,  p.  249. 

3  Pogg.  Ann.,  1865,  Vol.  125,  p.  177. 

4  Wiener  Sitzung.,  1860,  Vol.  40,  p.  607. 
6  Pogg.  Ann.,  1876,  Vol.  159,  p.  399. 


COEFFICIENT  OF  SLIP  IN  AIR.  241 

In  the  determination  of  the  elementary  electrical  charge  by  the  falling 
drop  method  Millikan  *  found  that  very  small  droplets  of  oil  did  not 
follow  Stokes'  equation 

(2)  v  =  2~^(°  ~  P)- 

From  his  observations  he  made  an  empirical  correction  of  the  above 
equation,  writing  the  corrected  law  in  the  form 

(3)      .         :    •-§?<.-. 

where  Al  was  determined  from  the  curve  of  his  observations.     Millikan 
pointed  out  that  the  correction  of  Stokes'  Law  for  Slip  z  gave 


which,  for  small  values  of  f/o,  reduces  to 

(5)  *=-^(<r-p)(i  +  r/a). 

9  1 

Thus,  the  Al  determined  by  Millikan  was  really  the  coefficient  of  slip 
for  oil  and  air.  His  value  for  23°  C  was  f76  =  77  X  io~7.  In  a  later 
determination  3  a  more  detailed  study  of  the  failure  of  Stokes'  Law  for 
small  oil  drops  gave  the  value  £76  at  23°  C  =  82.2  X  io~7. 

For  small  drops  of  shellac,  Lee  4  observed  Al  =  fr6  at  76  cm  and  23°  C 
to  be  100  X  io~7. 

These  experimental  results  led  Millikan  to  a  theoretical  study  of  slip 
for  different  boundary  conditions.  He  concluded  that  when  no  gas 
molecules  were  regularly  reflected  after  impact  upon  the  walls,  the 
maximum  of  external  friction,  and  hence  the  minimum  of  slip,  would 
result.  For  a  mechanically  rough  surface,  which  would  cause  such 
diffuse  reflection  of  all  gas  molecules,  he  calculated  the  minimum  value 
of  f  at  23°  C  and  76  cm  to  be  65.9  X  lO"7. 

III.  THEORY  FOR  SLIP  DETERMINATIONS  BY  THE  CONSTANT  DEFLECTION 

METHOD. 

The  theory  of  the  Constant  Deflection  Method  of  determining  viscosity 
coefficients  gives 


1  PHYS.  REV.,  1911,  32,  382. 

2  Bassett,  Hydrodynamics,  Vol.  II.,  p.  271. 
J  PHYS.  REV.,  1913,  Vol.  II.,  p.  139. 

4  PHYS.  REV.,  1914,  IV.,  420. 


242  LELAND   JOHNSON  STACY. 

I  =  moment  of  inertia  of  suspended  cylinder ;  <f>  —  angular  displacement 
of  suspended  cylinder;  a  =  radius  of  suspended  cylinder;  /  =  length  of 
suspended  cylinder;  T  =  period  of  oscillation  of  suspended  cylinder; 
co  =  angular  velocity  of  outer  cylinder;  b  =  radius  of  outer  cylinder, 
when  it  is  assumed  that  there  is  no  slip  at  the  surfaces  of  the  cylinders. 
At  ordinary  pressures,  this  assumption  involves  an  error  too  small  to  be 
observed  by  this  method.  For  the  case  where  slip  becomes  appreciable 
the  complete  equation  as  developed  by  Millikan  is 


(7)  u  =  -. 


a2\/      ,  63  +  a3   \ 

2     J\  ^Pab3  -asb) 


The  slip  term  at  atmospheric  pressure  may  be  calculated  for  this  appa- 
ratus since  a  =  5.3412  cm,  b  =  6.0632  cm,  and  f76  =  66  X  io~7  (from 
theory).  This  gives  a  value  for  the  term 

b3  +  a3 

2{P  -=~  —  -  =  .00002 
ab3  —  a3b 

which  is  quite  negligible,  since  the  experimental  error  is  about  .1  per  cent. 
Hereafter  the  equation  (6)  will  be  written 


and  tip   denned  as  the  apparent  viscosity  coefficient  at  the  pressure  p. 
The  value  1776'  will  be  taken  as  the  true  value  of  the  viscosity  coefficient. 
Equation  (7)  may  now  be  written 


and  solving  for  f  p 

(10)  fp  =  (-^-- 

Thus  to  determine  the  slip  coefficient  by  this  method  it  is  necessary, 
first,  to  determine  the  coefficient  of  viscosity  (77  =  T776')  and  then  ijp  at 
some  other  pressure.  Application  of  equation  (10)  gives  fp,  and  f76  is 
calculated  from  the  formula  f76  =  fp(£/76).  17'  and  77 p  must,  of  course, 
be  observed  at  the  same  temperature  or  reduced  to  the  same  conditions 
by  the  proper  formula. 

The  investigation  of  slip  coefficients  by  this  method  was  undertaken, 
at  Prof.  Millikan's  suggestion,  by  E.  L.  Harrington  x  to  determine 
whether  the  slip  depended  on  the  nature  of  the  surface  as  the  lack  of 

1  PHYS.  REV.,  1916,  VIII.,  p.  738. 


COEFFICIENT  OF  SLIP  IN  AIR. 


agreement  between  the  values  for  oil  and  shellac  drops  had  indicated. 
Harrington  devoted  his  time  chiefly,  however,  to  the  improvement  of 
the  precision  of  the  method  of  determining  the  viscosity  coefficient, 
leaving  the  writer  to  carry  on  the  slip  determinations.  The  writer 
assisted  Harrington  for  a  short  time  in  the  first  determinations  of  the 
slip  coefficient.  Some  of  Harrington's  results  will  be  in- 
cluded in  this  paper. 

IV.    ADAPTATION  OF  THE  APPARATUS  FOR  SLIP  DETER- 
MINATIONS. 

The  Constant  Deflection  Apparatus  consists  essen- 
tially of  two  concentric  brass  cylinders,  the  inner,  /, 
being  suspended  on  an  elastic  suspension,  s,  so  that, 
when  the  outer  cylinder,  O,  is  driven  at  a  constant  speed 
by  a  clock  driving  mechanism,  K,  a  constant  torque  due 
to  viscosity  will  cause  a  constant  deflection  of  the  sus- 
pended cylinder  from  its  equilibrium  position.  To  elim- 
inate end  effects  the  inner  cylinder  is  suspended  between 
two  guard  rings,  G,  of  the  same  diameter  and  less  than 
.3  mm  from  it.  A  small  mirror  mounted  at  the  base  of 
the  suspension  wire  makes  it  possible  to  observe  the  de- 


Fig,  i. 


flection  by  a  telescope  and  scale.  The  period  of  rotation  (t  =  27r/o>)  of 
the  outer  cylinder  was  determined  by  a  chronograph  attached  to  the 
driving  mechanism. 

The  constants  of  the  apparatus  as  determined  by  Harrington  were  used. 

Moment  of  inertia  of  inner  cylinder /  =  7617.3 

Radius  of  inner  cylinder a  =        5.3410  cm 

Length  of  inner  cylinder /  =      24.88      cm 

Radius  of  outer  cylinder b  =        6.0632  cm 

For  work  at  low  pressures,  the  cylinders  were  set  upon  a  steel  plate, 
P,  about  30  cm  in  diameter.  A  large-mouthed  glass  bottle,  /,  about 
28  cm  in  diameter  and  62  cm  high  was  found  and  the  mouth  ground 
plane  to  fit  tightly  a  rubber  gasket  laid  on  the  steel  base-plate.  An 
opening  was  ground  in  the  bottom  of  the  bottle  for  the  suspension  head, 
T,  which  projected  some  25  cm  above  the  large  bottle.  A  glass  tube,  A, 
sealed  at  the  top  was  fitted  by  a  ground  glass  joint  into  the  bottom  of  the 
large  bottle  thus  closing  the  apparatus.  A  plate  glass  observing  window 
was  sealed  over  a  small  hole  in  the  side  of  the  suspension  cover  to  prevent 
distortion  of  the  image  seen  in  the  observing  mirror.  A  discharge  tube 
for  spectroscopic  work  was  also  sealed  into  the  side  of  the  suspension  cover. 
The  driving  shaft  for  the  rotating  cylinder  was  led  through  an  iron  pipe, 


244  LELAND   JOHNSON  STACY. 

Q,  sealed  into  the  steel  base-plate.  The  lower  end  of  this  pipe  was 
immersed  in  a  vessel  of  mercury,  C,  thus  sealing  it  from  the  outside. 
A  rim  around  the  edge  of  the  base-plate  made  it  possible  to  use  a  mercury 
seal  at  this  joint  and,  the  bottom  of  the  bottle  being  somewhat  concave, 
a  mercury  seal  was  also  used  at  the  upper  joint.  Thus  the  apparatus 
was  enclosed  tightly  enough  to  permit  evacuation  to  about  .001  mm 
pressure. 

A  Gaede  mercury  pump  backed  by  a  rotary  oil  pump  made  it  possible 
to  reduce  the  pressure  within  the  apparatus  to  .1  mm  in  a  little  over 
two  hours.  The  volume  of  the  apparatus  was  about  150  liters.  Pres- 
sures were  read  by  a  McLeod  gauge  calibrated  to  read  .0001  mm.  Tem- 
peratures within  the  apparatus  were  read  from  a  Beckmann  thermometer 
which  had  been  calibrated  by  comparison  with  a  standard  Baudin 
instrument.  The  observing  telescope  and  scale  were  mounted  about 
200  cm  distant  from  the  suspended  mirror  and  the  deflection  could  be 
read  to  .1  mm.  The  steel  suspension  wire  used  throughout  this  work 
gave  an  observed  scale  deflection  of  62  to  64  cm  when  the  period  of  rota- 
tion of  the.outer  cylinder  was  about  30  sec. 

V.   EXPERIMENTAL  METHODS  AND  ELIMINATION  OF  ERRORS. 

The  determination  of  the  viscosity  coefficient  was  made  from  the 
formula 

/  -.-,-    t  ,    S  <  2?T  -    S 

rjp   =  K  —  tan"1 —         where         /  =  —  ;         $  =  tan"1  —  > 
T2  2#  co  2a 


The  period  T  and  the  scale  distance  d  being  determined  beforehand,  it 
was  only  necessary  to  observe  5  and  /  so  that  r)p'  could  be  calculated. 
Temperatures  and  pressures  were  read  as  already  described. 

Having  determined  f]^'(=  •>/),  it  was  necessary  to  determine  -rjp'  at 
some  low  pressure  and  calculate  fre  from  the  equations 


*:,\tt.!t         •.*: 

For  atmospheric  pressure  the  observed  scale  deflection  5  was  about  63 
cm  with  an  error  of  .3  mm.  At  0.12  mm  pressure  and  t  =  30  sec.  the 
scale  deflection  was  found  to  be  about  56  cm.  Thus  a  difference  of  7  cm 
with  an  error  of  .3  mm  in  reading  would  give  an  apparent  error  of  not 
more  than  I  part  in  200.  The  gauge  reading  to  .0001  mm,  this  error 
should  be  only  i  part  in  1,200.  Errors  in  observing  all  the  other  factors 
were  much  less  than  these,  so  they  may  be  neglected.  It  was  found, 


COEFFICIENT  OF  SLIP  IN  AIR.  245 

however,  that  the  principal  source  of  inaccuracy  was  due  to  change  of 
the  pressure  during  observations.  This  rise  of  pressure  was  considerable 
in  the  first  few  hours  after  evacuation  was  stopped,  but  reached  a  steady 
value  after  24  hours  or  so.  It  was  found  by  experiment  that  during  the 
lo-minute  interval  necessary  for  one  complete  observation  the  pressure 
change  was  from  .001  to  .003  mm.  After  one  or  two  days  the  pressure 
rose  less  than  that  in  24  hours.  This  was  explained  by  supposing  the 
increase  in  pressure  to  be  due  to  gases  released  from  the  glass  and  metal 
surfaces.  By  taking  pressure  readings  before  and  after  each  observation 
a  fairly  accurate  mean  value  was  obtained.  The  temperature  was  also 
read  at  frequent  intervals  and  the  mean  value  used.  The  apparatus 
being  set  up  in  a  constant  temperature  room,  the  observations  were  made 
within  a  very  narrow  range  (22°  to  24°  C).  Variations  from  23°  C  were 
corrected  for  by  Millikan's  formula  (Ann.  der  Physik,  1913,  Vol.  41,  p. 

759)- 

-no  =  >?23°  -  .000000493(23°  -  0). 

In  the  early  work  on  slip  determinations,  a  series  of  observations  was 
made  after  a  single  evacuation.  The  values  of  fre  calculated  from  these 
observations  gave  an  initial  value  of  about  70  X  io~7  but  rose  steadily 
until  a  value  of  200  X  io~7  was  found  about  two  weeks  later.  The 
pressure  change  was  from  .1238  to  .1448  mm  during  this  interval.  Since 
f7e  should  be  independent  of  the  pressure,  this  indicates  that  the  increase 
in  pressure  must  be  due  to  the  presence  of  some  gas  of  a  lower  viscosity 
than  air.  This  suggested  that  hydrogen  (viscosity  about  one  half  that 
of  air)  was  being  released  from  occlusion  by  the  metal  parts  of  the 
apparatus.  Spectroscopic  examination  of  the  discharge  tube  showed  a 
definite  increase  in  the  intensity  of  the  hydrogen  lines  when  the  appa- 
ratus was  allowed  to  stand  several  days  at  a  low  pressure. 

Admission  of  air  to  full  atmospheric  pressure  to  flush  out  the  apparatus 
and  a  second  evacuation  gave  the  same  result,  viz.,  a  low  value  during 
the  first  two  or  three  hours  after  the  pumps  were  stopped,  then  a  steady 
rise  in  the  value  of  the  slip  constant  fre.  To  eliminate  this  variation  it 
was  found  advisable  to  admit  air  immediately  after  a  set  of  observations 
was  completed  and  to  evacuate  only  a  short  time  before  readings  were 
to  be  taken.  Thus  the  time  during  which  the  "hydrogen  effect"  might 
be  present  was  so  short  that  it  did  not  affect  the  results  appreciably. 
It  was  found  that  the  values  of  f76  obtained  within  three  or  four  hours 
after  an  evacuation  were  quite  consistent.  Observations  were  usually 
made  within  an  hour  after  evacuation.  After  many  evacuations  this 
"hydrogen  effect"  was  less  marked  but  was  always  present.  By  taking 
observations  shortly  after  evacuation,  it  was  avoided. 


246 


LELAND   JOHNSON  STACY. 


VI.  TABLE  OF  OBSERVATIONS  AND  CALCULATED  DATA  FOR  BRASS 
SURFACES  IN  AIR. 

Results  on  Brass  Surfaces  in  Air. 
d  =  200.5  cm;  T  =  175.48;  11  =  1822.6  X  io~7. 


5  (cm). 

*  (sec.). 

e  (°  C). 

7?p  X  I07. 

rP  x  io". 

p  (mm). 

fn  x  io7. 

56.28.  .  .  . 
56.56.  .  .  . 

.  30.134 
30.040 

23.04 
23-05 

1643.0 

1646.0 

3884 
3811 

•1303 
.1328 

66.6 
66.6 

56.71---. 
56.93  -.. 

30.034 
29.989 

22.76 
22.81 

1649.4     3701 

l653.2      3616 

.1361 
.1388 

66.4 
66.0 

57-23.... 
55-95.... 

56.76.  .  .  . 
56.63.... 

29-500 
30.267 

29.992 
30.124 

22.67 
22.67 

22.72 
22.72 

1634.7      4044 
1640.2      3912 

1646.0      3726 
1646.2      3721 

.1238 
.1266 

.1340 
.1362 

65-9 
65-2 

65-7 
66.7 

55-77.... 
56.25.... 

30.187 
29-975 

22.81 
22.82 

1628.2      4166 
1630.6      4IO7 

.1200        65.8 
.1209        65.3 

57-25  •-.. 
57-02.... 

29.720 
29.880 

23-70 
23.66 

1645.0 
1647.1 

3856 

3797 

•1253 
.1268 

63-6 
63-4 

57-10.... 
56.68.... 

29-795 
30.032 

22.81 
22.81 

1644.8 
1645-9 

3764 
3738 

.1287 
-I3II 

63-8 
64-5 

56.71--.. 
56.86.  .  .  . 

30.134 
30.079 

22.80 
22.83 

1652.9 
1653.8 

3570 
3556 

.1361 
•1367 

63-9 
64.0 

57-20.  .  .  . 
57-27.-.. 

29.929 
29.922 

22.82 
22.79 

1655-1      3520 
1656.7      3480 

.1405 
.1428 

65-1 
65-4 

57-15.-.. 
57-05-... 

30.000 
30.053 

22.80 
22.73 

1657.6      3459 
1657-7      3451 

.1445        65.8 

.1459      66.2 

57.32-... 
57-30.  .  .  . 

29.922 
29.930 

22.79 
22.77 

1656.2 
1656.5 

3437 
3444 

.1467 
•1475 

66.3 
66.8 

56.51-... 

29.952 

22.87 

1640.5      3836 

.1368      69.0 

55.77.... 
55.65..-. 

29.917 
30.000 

22.84 
22.78 

1614.0 
1614.6 

4469 
4436 

.1178      69.3 
.1166      68.1 

56.47.... 

29.873 

22.90 

1631.2      4053 

.1300      69.3 

56.53-  •  •  • 

30.037 

22.42 

1641.9      3733 

.1402      68.9 

56.73-..  - 

30.043 

23-01 

•1651.4      3405 

.1508      67.6 

56.62...  . 

29-915 

22.73 

1637.0      3886 

.1343      68.7 

56.68.... 

30.102 

22.84 

1649.8      3591 

.1349      63.7 

56.46.  .  .  . 
56.76.  .  .  . 

30.100 
30.018 

22.81 
22.79 

1647.1      3646 
1647.4      3630 

.1378       66.1 
.1424      68.0 

56.68.... 
57.00.  .  .  . 

29.972 
29.860 

22.87 
22.99 

1642.6 
1645.6 

3762 
3710 

.1308      64.7 
.1338      65.3 

57-03.... 

30.158 

23-33 

1662.9 

3338 

•1558 

68.3 

56.34-  •  •  • 

30.080 

23-46 

1638.8 

3915 

.1258 

64.8 

Mean  .  .  . 

66.15 

COEFFICIENT  OF  SLIP  IN  AIR.  247 

The  observations  were  made  within  a  pressure  range  of  .1  to  .18  mm 
since  at  lower  pressures  the  error  due  to  change  of  pressure  during  an 
observation  was  large,  while  at  pressures  above  .18  mm  the  deflection 
5  differed  so  little  from  the  deflection  at  atmospheric  pressure  that 
the  error  of  observing  this  difference  became  considerable. 

In  general,  the  experimental  conditions  were  as  follows: 

Temperature  —  22°  C  to  24°  C 
Pressure          —  .1000  mm  to  .1800  mm 
Period  (t)        —  30  sec.  (±.2) 
Period  (r)       -  175.5  sec. 

Scale  deflection,  for  p  =  76       cm   -630  mm  1 

"  p  =    o.i 8  mm  -  580  mm  >•  approximately 
"  p  =    o.io  mm  -  550  mm  J 

Observations  were  usually  made  in  pairs  within  an  hour  after  evacua- 
tion. 

The  mean  value  (66.15  X  io~7)  is  very  close  to  Millikan's  theoretical 
minimum  value  (65.9  X  io7)  but  is  considerably  lower  than  any  of  the 
values  found  by  other  methods  for  oil  or  for  glass  surfaces. 

VII.   DETERMINATION  OF  SLIP  FOR  SHELLAC  SURFACES  IN  AIR. 

The  cylinders  were  next  coated  with  a  thin  layer  of  shellac,  dried  by 
an  air  blast  and  replaced  in  position.  From  the  weight  of  the  cylinders 
before  and  after  the  shellac  was  applied  and  the  surface  area  of  the 
cylinders,  the  average  thickness  of  the  shellac  film  was  calculated.  This 
was  found  to  be  .01  mm  and  made  only  a  small  correction  in  the  values 
of  the  constants  a  and  b. 

After  the  determination  of  the  viscosity  coefficient  at  atmospheric 
pressure,  the  pressure  was  reduced  and  observations  made  as  before. 
At  first  a  high  value  of  fre  =  97  X  io~7  was  found  while  the  shellac  was 
fresh.  Later,  the  values  of  fa  dropped  steadily  toward  the  minimum 
value.  This  result  is  shown  in  the  two  sets  of  data  here  given ;  the  first 
set  of  observations  being  due  to  Harrington  and  the  second  to  the  writer. 
In  two  other  trials,  accidental  experimental  difficulties  prevented  the 
writer  from  making  observations  while  the  shellac  was  fresh  but  values 
between  80  and  90  X  io~7  were  found  several  days  after  application 
of  the  shellac.  The  fall  in  the  observed  value  of  the  slip  coefficient 
indicates  a  change  in  the  surface  due  probably  to  oxidation  of  the 
shellac  which  seems  to  produce  a  rough  surface. 


248 


LELAND   JOHNSON  STACY. 


Data  on  Fresh  Shellac  Surfaces  in  Air. 

d  =  200.6,  T  =  176.00. 
E.  L.  Harrington:  Fresh  Shellac,  August  i,  ipi6. 


Date. 

S. 

t. 

e. 

ij/  X  io7. 

TP  X  IOB. 

p  (mm). 

ft.  X  io7. 

Aug.  9  

55-93 

29.719 

23-38 

1602.8 

4911 

.1474 

95-2 

55-72 

29.727 

23-44 

1597-3 

5059 

.1478 

98.5 

Aug.  10,  A.M.  . 

53-42 

29-715 

23.76 

I53L3 

6842 

.1048 

94-3 

52-55 

29.960 

23-86 

I5I9-4 

7188 

.1058 

IOO.O 

Aug.  10,  P.M.. 

53-99 

29.936 

23.67 

1559-0 

6080 

.1125 

90.0 

53-19 

30.263 

23.60 

1552.9 

6240 

•1133 

93-0 

Aug.  ii,  A.M.. 

53-43 

30.227 

23.46 

1556.6 

6121 

.1107 

89.0 

52.96 

30.440 

23-50 

1553-9 

6204      .1112 

90.8 

Aug.  ii,  P.M.. 

55-19 

30.100 

23-13 

1602.0 

4886 

.1288 

82.8 

55-H 

30.100 

23-13 

1600.5 

4920 

.1298 

84.1 

Aug.  12  

56-94 

29.827 

23.64 

1637.0 

4099 

.1438 

77-6 

56.75 

29.917 

23.70 

1636.5 

4108 

.1438 

77-7 

Aug.  30  

57-28 

29.780 

23.10 

1653-1 

3538 

.1506 

70.1 

L.  J.  Stacy:  Fresh  Shellac,  March  7,  1917. 


Mar.  9,  A.M.. 

56.04 

29.789 

23-36 

1578.8 

5588 

.1330 

97-7 

55-92 

30.010 

23.40 

1582.5 

5381 

•1352 

Mar.  9,  P.M.. 

57-66 

29.800 

23-14 

1619.7 

4580 

.1470 

88.6 

57-26 

30.080 

23.19 

1623.7 

4486 

.1482 

87.5 

Mar.  io,  A.M.. 

58.15 

29-723 

23-55 

1628.2 

4414 

.1418 

82.3 

57-76 

29.983 

23-59 

1632.4 

4318 

.1441 

81.9 

Mar.  io,  P.M.. 

57.56 

30.164 

23-53 

1636.7 

4206 

•1493 

82.6 

58.12 

29.896 

23-50 

1637.7 

4179 

•1543 

84.8 

57-30 

30.322 

23.24 

1637-9 

4152 

•1545 

84.4 

Mar.  ii  

56.83 

29.985 

22.87 

1  606.6 

4876 

•I3I3 

84.2 

56.06 

30-432 

22.86 

1608.7 

4820 

•1323 

83-9 

Mar.  12  

57.38 

30.334 

23.12 

1640.8 

4066 

.1487 

79-5 

57-77 

30.188 

23.21 

1643.8 

3999 

•1495 

78.7 

Mar.  13  

57-75 

30-144 

23-52 

1640.9 

4102    .1443 

77-9 

58-05 

30.031 

23-52 

1643.1 

4048    .1473 

78.4 

Mar.  16  

58-16 

30.169 

23.07 

1653-7 

3747 

.1518 

74.8 

Mar.  17  

57-42 

30.084 

22.85 

1628.4 

4242 

.1332 

74-3 

The  following  results  were  obtained  using  the  same  experimental 
method  as  with  brass  surfaces.  The  shellac  surfaces  were  about  two 
months  old  when  the  first  readings  were  taken. 

The  results  of  these  experiments  furnish  independent  evidence  of  the 
fact  that  the  viscosity  of  a  gas  is  independent  of  the  pressure,  since  foe 
turns  out  to  be  a  constant  for  a  given  surface  in  air.  The  value  of  fre 
for  rough  surfaces  checks  Millikan's  theoretically  deduced  values  within 
the  limit  of  error  of  the  experiment.  The  variation  in  slip  for  different 
surfaces  has  been  checked  and  the  result  for  fresh  shellac,  f76  =  96.8X  io~7, 
is  very  close  to  the  result  Lee  obtained,  fre  =  100  X  io~7,  from  the 
correction  of  Stokes'  Law  for  falling  shellac  drops. 

In  conclusion  the  writer  wishes  to  express  his  indebtedness  to  Professor 
R.  A.  Millikan  who  suggested  the  problem  and  directed  the  experimental 


COEFFICIENT  OF  SLIP  IN  AIR. 


249 


Data  for  Old  Shellac  Surfaces  in  Air. 
d  =  200.6  cm;  T  =  176.00  sec. 


5. 

t. 

e. 

r,p'  X  I07. 

f,  X  10 

P  (mm). 

fte  X  io7. 

55-39  .... 

30.027 

23.67 

1605.1 

4833 

.1088 

69.2 

57.49.  .  .  . 
57.06  

29.767 
30.076 

24.28 
24-34 

1648.5 
1653-3 

3875 
3767 

.1338 
.1370 

68.2 
67.9 

5749.  •  •  • 

30.023 

23.11 

1661.5 

3443 

•1535       69.5 

57.02.... 
57.20.  .  .  . 

30.058 
30.123 

22.76 
22.87 

1653.8 
1662.5 

3639 
3431 

.1444 
.1478 

69.1 
66.7 

57.38-  •  •  • 
57-28.... 

30.023 
30.074 

22.90 
22.95 

1661.7 
I662.I 

3447 
3444 

.1488       67.5 
.1493       67.7 

57.64-  •  •  - 

57-72.  .  .  . 

29.885 
29.930 

22.92 
22.97 

1659.2 
1664.4 

3507 
3392 

.1478       68.2 
.1495       66.7 

57-44.... 
57-60.... 

30.144 
30.050 

23.08 
23.11 

1668.9 
1668.2 

3299 
3317 

•1573       68.3 
.1585       69.2 

56.75  •..• 
56.90.  .  .  . 

30.111 
30.112 

23.24 
23.28 

1647.3 
1651.1 

3830 
3744 

•1351 
.1373 

68.1 
67.6 

57-I3--  •• 

30.067 

23.21 

1656.0 

3613 

•1437       68.3 

55.19..-. 
55.23...- 

57.15.... 
57.27.... 

30.014 
30.087 

30.210 
30.271 

23.06 
23.10 

23-57 
23.66 

1597-2 
1602.3 

1664.0 
1670.8 

5032 
4910 

3464 
33i6 

•1057 
.1076 

.1498 
•1533 

70.0 

69-5 

68.3 
66.9 

56.91..-. 
57.I7-... 

30.211 
30.174 

23-21 

23-31 

1657.2 
1662.6 

3585 
3469 

.1411       66.6 
.1445       66.0 

57.36.  .  .  . 
57.37...- 

29.963 
30.019 

22.55 
22.58 

1650.5 
1652.8 

3823 
3793 

•1353 
.1380 

68.1 
68.6 

56.74.  .  .  . 

30.300 

22.51 

1644.5 

3780 

.1288 

64-1 

57.33..-. 
57.25...- 

30.161 
30.160 

22.87 

22.86 

1655.3 
1652.6 

3742 
3806 

•1356 
.1364 

66.8 
68.3 

58.38.-.- 
57.23..-. 

29.508 
30-152 

21.94 

22.12 

1647-3 
1655.5 

3835 

3777 

•1363 

-1373 

68.8 
68.2 

Mean  .  .  . 

67.7 

This  result  is  2\  per  cent  higher  than  the  value  found  for  brass  surfaces  in  air. 

work ;  to  the  other  members  of  the  Physics  Department  of  the  University 
of  Chicago  for  their  interest  and  assistance  throughout  the  investigation ; 
and,  in  particular,   to   Dr.   E.   L.   Harrington  with  whom  the  author 
worked  in  the  first  determination  of  slip  by  this  method. 
RYERSON  PHYSICAL  LABORATORY, 
UNIVERSITY  OF  CHICAGO, 
September  22,  I922.1 

*The  work  described  in  this  paper  was  completed  in  February  1919. 


Pftys 
DATE 

!'ca'  Sciences 
DUE 

M&raiy 

Phys.Sci. 
>•    QC189       Harrington,   E. 

37J 

H37 
A  redetermination  of  the 

coefficient  of  viscosity 

of  air. 

1  1  III!   LI  1  II    II    1  II    1 

lillH 

3  W*JC^§ee524221 

Gty 

Phys  .  Sci  • 

QC189      Harrington,   E» 
-  H37 

A  redetermination  of  the 

GAYLORD 

PR.NTEO.NU.S.A. 

coefficient  of  viscosity 
of  air» 

3ATEtOANED|                                  ISSUED  TO                               }     DATE  DUE 

Physical  Sciences  Library 

University  of  California 

Riverside 


TIP 


eprinted  from  NATURE,  Vol.  136,  page  682,  October  26,  1935.) 


Viscosity  of  Air  and  the  Electronic  Charge 
' 

THE  greatest  uncertainty  in  determining  the 
electronic  charge  e  by  the  oil  drop  method  of 
Millikan  is  introduced  by  the  uncertainty  in  the 
assumed  value  of  the  coefficient  of  viscosity  of  air,  TJ. 
The  value  adopted  by  Millikan  in  1917 

T]23°  =  (1822-6±l-2)  x  10-7 

is  probably  too  low,  and  its  accuracy  overestimated, 
as  is  pointed  out  by  Shiba1. 

Considering  the  fundamental  importance  of  the 
constant  e,  I  have  undertaken  a  new  determination 
of  v),  using  the  rotating  cylinder  method  also  em- 
ployed by  Millikan  and  his  co-workers2>3  :  An  inner 
cylinder  of  electron  metal,  suspended  vertically  by 
a  fine  phosphor-bronze  wire  between  two  guard 
cylinders  of  equal  diameter  is  deviated  from  its 
equilibrium  position  through  an  angle  9  by  a  con- 
centric outer  cylinder,  rotating  with  constant  velocity, 
Y)  being  calculated  from  the  equation 

/  (ft2  -  a2). 9  * 
73  =  2  a*6*  Z     T2      '    where 
a  =  the  radius  of  the  inner  cylinder  =  2-81767  cm. 

at  20° ; 

6  =   the  radius  of  the  outer  cylinder  =  3-26628  cm. 
at  20°  or  =  3-18328  cm.  at  20°  (two  different 
cylinders) ; 
I    =   the  length  of  the  inner  cylinder  =  9-9981  cm. 

at  20°; 
t    =   the  time  of  revolution  of  the  outer  cylinder 

(20-150  sec.); 

T  =  the  period  of  oscillation  of  the  suspended  system 

(53  —  128    sec.,   using   different   suspensions)  ; 

/  =   the  moment  of  inertia  of  the  suspended  system 

about  the  line  of  suspension  =  423-22  gm.cm2. 

The  mean  value  of  51  determinations  of  TJ  for  dry 
air  at  temperatures  between  18-9°  and  20-9°  is 

7] 20°  =  (1820-0  ±  3-0)  x  10-7  corresponding  to 
7]23°  =  (1834-8  ±  3-0)  x  10-7. 

From  this  we  get 

/1834-8\3/2 
e  =  (ig22^6J       x  4'770  x  10-10  =  (4-818  ±  0-012)  x 

the  uncertainty  stated  being  due  only  to  the  viscosity, 
other  sources  of  error  not  being  considered  here. 
I  am,  therefore,  of  the  opinion  that  the  discrepancy 


